3.89 \(\int x^{-1+k} (a+b x^k)^n \, dx\)

Optimal. Leaf size=23 \[ \frac{\left (a+b x^k\right )^{n+1}}{b k (n+1)} \]

[Out]

(a + b*x^k)^(1 + n)/(b*k*(1 + n))

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Rubi [A]  time = 0.0077646, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ \frac{\left (a+b x^k\right )^{n+1}}{b k (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + k)*(a + b*x^k)^n,x]

[Out]

(a + b*x^k)^(1 + n)/(b*k*(1 + n))

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int x^{-1+k} \left (a+b x^k\right )^n \, dx &=\frac{\left (a+b x^k\right )^{1+n}}{b k (1+n)}\\ \end{align*}

Mathematica [A]  time = 0.0100025, size = 23, normalized size = 1. \[ \frac{\left (a+b x^k\right )^{n+1}}{b k (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + k)*(a + b*x^k)^n,x]

[Out]

(a + b*x^k)^(1 + n)/(b*k*(1 + n))

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Maple [A]  time = 0.043, size = 29, normalized size = 1.3 \begin{align*}{\frac{ \left ( a+b{x}^{k} \right ) \left ( a+b{x}^{k} \right ) ^{n}}{b \left ( 1+n \right ) k}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+k)*(a+b*x^k)^n,x)

[Out]

(a+b*x^k)/b/(1+n)/k*(a+b*x^k)^n

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+k)*(a+b*x^k)^n,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.93785, size = 55, normalized size = 2.39 \begin{align*} \frac{{\left (b x^{k} + a\right )}{\left (b x^{k} + a\right )}^{n}}{b k n + b k} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+k)*(a+b*x^k)^n,x, algorithm="fricas")

[Out]

(b*x^k + a)*(b*x^k + a)^n/(b*k*n + b*k)

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Sympy [A]  time = 55.2727, size = 75, normalized size = 3.26 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \wedge k = 0 \wedge n = -1 \\\frac{a^{n} x^{k}}{k} & \text{for}\: b = 0 \\\left (a + b\right )^{n} \log{\left (x \right )} & \text{for}\: k = 0 \\\frac{\log{\left (\frac{a}{b} + x^{k} \right )}}{b k} & \text{for}\: n = -1 \\\frac{a \left (a + b x^{k}\right )^{n}}{b k n + b k} + \frac{b x^{k} \left (a + b x^{k}\right )^{n}}{b k n + b k} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+k)*(a+b*x**k)**n,x)

[Out]

Piecewise((log(x)/a, Eq(b, 0) & Eq(k, 0) & Eq(n, -1)), (a**n*x**k/k, Eq(b, 0)), ((a + b)**n*log(x), Eq(k, 0)),
(log(a/b + x**k)/(b*k), Eq(n, -1)), (a*(a + b*x**k)**n/(b*k*n + b*k) + b*x**k*(a + b*x**k)**n/(b*k*n + b*k),
True))

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Giac [A]  time = 1.05887, size = 31, normalized size = 1.35 \begin{align*} \frac{{\left (b x^{k} + a\right )}^{n + 1}}{b k{\left (n + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+k)*(a+b*x^k)^n,x, algorithm="giac")

[Out]

(b*x^k + a)^(n + 1)/(b*k*(n + 1))